For now I will focus on the conceptual, pending feedback from you:
What I'm interested in to know if it is possible to take two adjacent
beams, and do another step of beamforming,
The answer to this is yes, this is possible to do, and this is done in many beamforming applications. Think about it like this: All beamforming is really doing, is undoing the delays your signals exhibit (by virtue of different sensors at different locations), and summing the result. (Example: Delay-And-Sum beamformer). But in what order you undo the delays, before summing doesn't matter.
Thus, the beamformed result of a sub-group of sensors can be considered as one 'element', and the beamformed result of another sub-group of sensors, another 'element', and then both 'element's' outputs are beamformed.
In other words, first we remove the effects of the delays in one sub-array. Then we remove the effects of the delays in the other sub-array. At this point, both signal outputs might still have a delay relative to each other. So then again, both those inputs can be beamformed again. You can keep doing this ad infinitum.
The objective is to reduce my side lobes even further, and to get
narrower main beam.
If this is the objective, then the only way to reduce your main lobe width is to sample the spatial field with more sensors. This is the direct analogue to time-domain sampling. If we want greater frequency resolution (smaller main lobe width in the DFT analysis), we need to process a greater number of samples in time. Similarly, if we would like a smaller main-lobe width of our spatial spectra, we need to sample the spatial field with more sensors.
EDIT: Taking from the above, if you want to decrease your mainlobe width, you need more spatial samples. However if you want to reduce your sidelobes, then you would use a variety of windowing/shading functons. (Each of those functions will give you a slight increase in mainlobe width, but that is the price to pay). I am not sure what shading you have used for the above plots, but that will be good to know.